Below is a problem I did. However, it did not match the back of the book. I would like to know where

anginih86 2022-06-22 Answered
Below is a problem I did. However, it did not match the back of the book. I would like to know where I went wrong.
Problem:
Solve the following differential equation.
y = y x x
Answer:
d y d x = y x x x = y x 1 y = x v d y d x = x d v d x + v x d v d x + v = v 1 x d v d x = 1 d v = d x x v = ln x + c y x = ln | x | + c y = x ln | x | + c x
The book's answer is:
y = x ln | k x |
Where did I go wrong?
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Answers (2)

candelo6a
Answered 2022-06-23 Author has 24 answers
You didn't do anything wrong. Your book just took the constant c to be ln | k | .
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Abram Boyd
Answered 2022-06-24 Author has 5 answers
y = x ln | x | + c x
= x ln | 1 x | + c x
= x ( ln | 1 x | + ln k ), put c = ln k
y = x ( ln | k x | ), you did everything right
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